Up until now, there has been developed an optical modulator such as a traveling wave electrode type of lithium niobate optical modulator (hereinafter simply referred to as an LN optical modulator) comprising a substrate (hereinafter simply referred to as an LN substrate) made of a material such as lithium niobate (LiNbO3) having an electro-optic effect to cause a refractive index of an incident light to be varied in response to an electric field applied to the substrate, thereby making it possible to form an optical waveguide and a traveling wave electrode in and on the substrate. The LN optical modulator can be applied to a large volume optical transmission system having a capacity in the range of 2.5 Gbit/s to 10 Gbit/s due to the excellent chirping characteristics. In recent years, the LN optical modulator thus constructed is under review to be applied to the optical transmission system having a super large capacity of 40 Gbit/s and therefore expected as a key device in this technological field.
(First Prior Art)
There are two types of LN optical modulators with respect to the states of the substrate, one substrate having a z-cut state, and the other having an x-cut state (or a y-cut state). Here, x-cut LN substrate type LN optical modulator will be described as the first prior art, wherein the LN optical modulator comprises an x-cut LN substrate and a coplanar waveguide (CPW) forming a traveling wave electrode. FIG. 13 is a perspective view showing the x-cut substrate type LN optical modulator. FIG. 14 is a sectional view taken along the line A-A′ of FIG. 13.
The conventional LN modulation device comprises an x-cut LN substrate 1, an SiO2 buffer layer 2, and an optical waveguide 3 formed to be flush with an upper surface of the x-cut LN substrate 1 wherein the SiO2 buffer layer 2 has a thickness of 200 nm to 1 μm and transparent to incident light having optical wavelength typically utilized for optical communications such as for example 1.3 μm and 1.55 μm. The optical waveguide 3 is formed with a process of evaporating a metal Ti (titanium) on the x-cut LN substrate 1 and a process of thermal diffusing at a temperature of 1050° C. for approximately 10 hours, the optical waveguide 3 forming a Mach-Zehnder interferometer (a Mach-Zehnder optical waveguide). The optical waveguide 3 includes two interaction optical waveguides, that is, two arms 3a and 3b at the position where an electric signal and an incident light are interacted with each other (the position will be referred to as an interaction portion). The conventional LN optical modulator further comprises a traveling wave electrode 4 having two ground electrodes 4b, 4c and a center electrode 4a placed between the two ground electrodes 4b, 4c. 
As shown in FIG. 14, the interaction optical waveguides 3a and 3b have widths represented by Wa and Wb, respectively. In this first prior art, the interaction optical waveguides 3a and 3b have same widths with each other, that is, Wa=Wb, where Wa and Wb both represent the same value exemplified by 9 μm. The legend Gwg represents the distance between the interaction optical waveguides 3a and 3b (or a gap between the waveguides), the distance being set at, such as, 16 μm. The legend Δ represents a distance in the horizontal direction between one edge of the center electrode 4a and the center of the interaction optical waveguide 3b (that is, the center line), the edge of the center electrode 4a facing the ground electrode 4c. In general, the interaction optical waveguides 3a and 3b are symmetrically positioned with respect to the center electrode 4a and the ground electrodes 4b and 4c, which leads to the fact that the distance in the horizontal direction between another edge of the center electrode 4a facing the ground electrode 4b and the center of the interaction optical waveguide 3b is represented by Δ.
In FIG. 14, the center of the center electrode 4a in the width direction is represented by a center line 18. The centers of the interaction optical waveguides 3a and 3b in the width direction are represented by center lines 19a and 19b, respectively.
FIG. 15 is a top view showing the optical waveguide 3. The length of the interaction optical waveguides 3a and 3b is represented by the legend L. The position of the line A-A′ in FIG. 15 corresponds to the position of the line A-A′ in the perspective view shown in FIG. 13, though only the optical waveguide 3 is shown in FIG. 15.
In this first prior art, both a bias voltage (generally a DC bias voltage) and a high frequency electric signal (an RF electric signal) are superimposed and applied between the center electrode 4a and the ground electrodes 4b and 4c, thereby resulting in the fact that the incident light is phase modulated not only by the RF electric signal but also by the DC bias voltage at each of the interaction optical waveguides. The buffer layer 2 is important in that it functions to expand a modulation bandwidth of the optical modulator by reducing a microwave equivalent refractive index nm of the electric signal traveling through the traveling wave electrode 4 to be close to an effective refractive index n0 of the incident lights traveling through the respective interaction optical waveguides 3a and 3b. 
The operation of the LN optical modulator thus constructed will be described hereinafter. Firstly, the DC bias voltage and the RF electric signal are necessary to be applied between the center electrode 4a and the ground electrodes 4b and 4c to realize the operation of the LN optical modulator.
FIG. 16 is a graph showing the relationship between the applied voltage and the output light power of the LN optical modulator under a certain condition with the DC bias voltage set at “Vb”. As shown in FIG. 16, the DC bias voltage “Vb” is generally set such that the output light power becomes middle value of the peak-to-peak value.
FIG. 17 is a graph showing the relationship between the distance A and the product of the half wavelength voltage Vπ and the length L (that is, Vπ·L, which is utilized as a barometer of the driving voltage), where the legend Δ represents the distance in the horizontal direction between one edge of the center electrode 4a and the center line 19b of the interaction optical waveguide 3b, and the legend L represents the length of the interaction optical waveguides. In this calculation, the value Δ is varied by changing the gap Gwg representing the distance between the interaction optical waveguides 3a and 3b. As shown in FIG. 17, the distance Δ in the horizontal direction between one edge of the center electrode 4a and the center line 19b of the interaction optical waveguide 3b should be small to a certain extent, and there exists an optimum value.
When the value Δ, the distance in the horizontal direction between the edge of the center electrode 4a and the center line 19b (19a) of the interaction optical waveguide 3b (3a), is set to be smaller in order to lower the driving voltage, the gap Gwg, the distance between the interaction optical waveguides 3a and 3b, also becomes smaller. The optical modulator, however, encounters such a problem that the optical power ratio between the condition of the optical output is ON state and OFF state, that is, extinction ratio, is deteriorated under the condition that the gap Gwg between the interaction optical waveguides 3a and 3b becomes smaller. This results from the fact that the degree of coupling between the interaction optical waveguides 3a and 3b becomes severely large.
(Second Prior Art)
There has been two methods to make the degree of coupling between the interaction optical waveguides 3a and 3b smaller, one being achieved by setting the gap Gwg wider so that the incident lights in the interaction optical waveguides 3a and 3b are transmitted to be away from each other, the other being achieved by setting the widths of the interaction optical waveguides different from each other so that the incident lights respectively traveling the incident optical waveguides have the respective effective refractive indexes (propagation constants) different from each other.
However, the value Vπ·L becomes larger as the gap Gwg between the interaction optical waveguides 3a and 3b becomes larger as shown in FIG. 17, which results in the fact that the driving voltage is required to be higher. To avoid this problem, the widths Wa′and Wb′ of the interaction optical waveguides 5a and 5b are set to be different from each other according to the second prior art as shown in FIG. 19. FIG. 20 is a top view showing the optical waveguide 5 according to the second prior art. FIG. 19 is a sectional view taken along the line B-B′ of FIG. 20, shown with the x-cut LN substrate 1, the center electrode 4a, the ground electrodes 4b and 4c, and the buffer layer 2.
The legend Δ′ represents the distance in the horizontal direction between one edge of the center electrode 4a and the center line of the interaction optical waveguide 5b, the edge facing the ground electrode 4c. In FIG. 19, the centers of the interaction optical waveguides 5a and 5b in the width direction are represented by center lines 20a and 20b, respectively.
The second prior art as described above, however, encounters some problems. Firstly, each of the widths Wa′ and Wb′ of the interaction optical waveguides 5a and 5b is set to be partially varied as shown in FIG. 20. Here, the interaction optical waveguides 5a and 5b form a first region having a length of L1 and a second region having a length of L2.
As shown in FIG. 20, the interaction optical waveguides 5a and 5b have taper portions 6, 7, 8, 9, 10 and 11 to have the widths of the interaction optical waveguides varied, each of the interaction optical waveguides 5a and 5b being required to have three taper portions. It is well known that radiation loss is caused at the portion where the width of the optical waveguide is varied. Furthermore, the radiation loss at the taper portion where the optical waveguide is widened and the radiation loss at the taper portion where the optical waveguide is narrowed are different from each other. Therefore, the incident lights respectively traveling the interaction optical waveguides 5a and 5b have powers different from each other, which results in the deterioration of the extinction ratio.
The major problem, that is, the chirping problem encountered by the second prior art will be described hereinafter. The degree of chirping can be represented by an alpha parameter (i.e., “a” parameter) as described by the formula (1), wherein the alpha parameter is calculated by a phase “φ” of the optical signal pulse outputted from the optical modulator and an intensity (amplitude) “E” of the optical signal pulse (disclosed in non-patent document 1).α=[dφ/dt]/[(1/E)(dE/dt)]  (1)
As can be seen in the above, the “α” parameter is calculated with an amount of phase shift and an amount of intensity variation of the optical signal pulse outputted from the optical modulator.
The “α” parameter can be represented by a formula (2), which is further developed from the formula (1).α=(Γ1−Γ2)/(Γ1+Γ2)  (2)
“Γ1”: An interaction efficiency normalized by the numerical number 1 in the form of overlap integration between the amplitude of the electric signal and the power of the incident light passing through the interaction optical waveguide 5a. 
“Γ2”: An interaction efficiency normalized by the numerical number 1 in the form of overlap integration between the amplitude of the electric signal and the power of the incident light passing through the interaction optical waveguide 5b. The value “Γ1” for the interaction optical waveguide 5a at the first region becomes equal to the value “Γ2” for the interaction optical waveguide 5b at the second region while the value “Γ2” for the interaction optical waveguide 5b at the first region becomes equal to the value “Γ1” for the interaction optical waveguide 5a at the second region, under the condition that the width Wa′ of the interaction optical waveguide 5a at the first region having a length of L1 is set to be equal to the width Wb′ of the interaction optical waveguide 5b at the second region having a length of L2 while the width Wb′ of the interaction optical waveguide 5b at the first region is set to be equal to the width Wa′ of the interaction optical waveguide 5a at the second region. However, this does not mean that the chirping becomes zero, that is, α=0, by setting the length L1 of the first region equal to the length L2 of the second region.
This stems from the fact that the traveling wave electrode 4 constituted by the center electrode 4a and ground electrodes 4b, 4c not shown in FIG. 20 causes high propagation loss to the high frequency electric signal traveling therethrough, which results in the high frequency electric signal attenuated as the high frequency electric signal propagates the traveling wave electrode 4. In order to make the alpha parameter “α” in the formula (2) to be zero, it is required to fulfill the following condition due to the attenuation of the high frequency electric signal.L1<L2  (3)
The condition to be imposed on the length L1 of the first region and the length L2 of the second region shown in FIG. 20 to make the chirping zero, that is, α=0 will be described hereinafter in detail. The microwave propagation loss at the frequency “f” under the condition that the high frequency electric signal is imposed on the traveling wave electrode 4 formed by the center electrode 4a and ground electrodes 4b, 4c is represented by βm(f). The integration values calculated by the interaction efficiency between the incident lights in the respective interaction optical waveguides 5a, 5b and the electric signal integrated by the length at the first region with the length of L1 and the second region with the length of L2 are represented by I1(f) and I2(f), respectively (the integration value simply be referred to as a modulation efficiency).
Each of the modulation efficiency I1(f) and I2(f) depends on the frequency “f” and can be described as the formulas (4) and (5) where the incident light and the electric signal propagate in the “z” direction.
                                          I            1                    ⁡                      (            f            )                          =                                            ∫              0                              L                ⁢                                                                  ⁢                1                                      ⁢                                          exp                ⁡                                  (                                                            -                                                                        β                          m                                                ⁡                                                  (                          f                          )                                                                                      ·                    z                                    )                                            ⁢                              ⅆ                z                                              ⁢                                          ⁢                                          =                                    (                              1                -                                  exp                  ⁡                                      (                                                                  -                                                                              β                            m                                                    ⁡                                                      (                            f                            )                                                                                              ·                                              L                        1                                                              )                                                              )                        /                                          β                m                            ⁡                              (                f                )                                                                        (        4        )                                                      I            2                    ⁡                      (            f            )                          =                                            ∫                              L                ⁢                                                                  ⁢                1                                            L                ⁢                                                                  ⁢                2                                      ⁢                                          exp                ⁡                                  (                                                            -                                                                        β                          m                                                ⁡                                                  (                          f                          )                                                                                      ·                    z                                    )                                            ⁢                              ⅆ                z                                              ⁢                                          ⁢                                          =                                    exp              ⁡                              (                                                      -                                                                  β                        m                                            ⁡                                              (                        f                        )                                                                              ·                                      L                    1                                                  )                                      ·                                          (                                  1                  -                                      exp                    ⁡                                          (                                                                        -                                                                                    β                              m                                                        ⁡                                                          (                              f                              )                                                                                                      ·                                                  L                          2                                                                    )                                                                      )                            /                                                β                  m                                ⁡                                  (                  f                  )                                                                                        (        5        )            
The chirping can be zero at any frequency “f” by setting the length L1 of the first region and the length L2 of the second region in such a way that the modulation efficiencies I1(f) and I2(f) fulfill the following condition.I1(f)=I2(f)  (6)
In other words, the alpha parameter becomes zero when the condition of the formula (6) is fulfilled.
According to the above described calculation, there is a relationship between the length L1 of the first region and the length L2 of the second region as follows.L1/L2≈0.9  (7)
However, manufacturing variations generally occurs typically in the width and the thickness of the center electrode 4a, the shape of the trapezoid and the inverted trapezoid, and the gaps between the center electrode 4a and ground electrodes 4b and 4c, due to the fact that the traveling wave electrode 4 is formed with a thick gold plating having a thickness of 20 μm or more. This results in the fact that the microwave propagation loss βm(f) at the frequency “f” of the high frequency electric signal propagating the traveling wave electrode 4 varies within the z-cut LN substrate not shown in FIG. 20, and varies from run-to-run of the manufacturing. Therefore, the process yield achieving the condition of α=0 can not be expected.
(Non-patent Document 1)
Nadege Courjal et al “Modeling and Optimization of Low Chirp LiNbO3 Mach-Zehnder Modulators With an Inverted Ferroelectric Domain Section “Journal of Lightwave Technology vol. 22 No. 5 May 2004